Algebra 2 / Unit 8: Trigonometric Functions
8.5 Modeling with Sinusoids
Notes
Repeating patterns like tides, Ferris wheels and daylight hours can be modeled with sine or cosine.
- Midline $\displaystyle k = \frac{\text{max} + \text{min}}{2}$. Amplitude $\displaystyle a = \frac{\text{max} - \text{min}}{2}$.
- $\displaystyle b = \frac{2\pi}{\text{period}}$.
- Start at a max: use $a\cos$. Start at a min: use $-a\cos$. Start at the midline going up: use $a\sin$.
- Example: high tide 10 ft, low tide 2 ft, period 12 hours: midline 6, amplitude 4, $\displaystyle b = \frac{\pi}{6}$.
Practice
A Ferris wheel 50 ft across has its center 30 ft above the ground and turns once every 40 seconds. A rider starts at the bottom. Which models the rider's height?
Amplitude is the radius 25, $\displaystyle b = \frac{2\pi}{40} = \frac{\pi}{20}$, and starting at the bottom means $-\cos$.
The water at a dock is 9 ft deep at high tide and 1 ft deep at low tide. What are the amplitude and midline?
Amplitude $\displaystyle \frac{9 - 1}{2} = 4$; midline $\displaystyle \frac{9 + 1}{2} = 5$.
A sinusoidal model has a period of 12 hours. What is $b$?
$\displaystyle b = \frac{2\pi}{12} = \frac{\pi}{6}$.
Ready for more? Take the Unit 8 test.